I have been reading Pippenger and Spencer's paper "Asymptotic behavior of the chromatic index for hypergraphs" and they comment that their result is applicable to the family of random k-uniform hypergraphs $G_k(n,p)$ whenever $p(n) = o(1)$ and $\frac{\log n}{n^{k-1}} = o(p(n))$.

I'm looking for a proof of this.

Let $d(G)$ and $D(G)$ denote the minimum and maximum degree of $G$ respectively. Let $C(G)$ denote the maximum co-degree in $G$: given $x,y \in V$, then $cod(x,y) = | \{ e \in E : x,y \in e \} |$. Basically, their theorem says that if

$d(G) \sim D(G)$ and $C(G) = o(D(G))$

then $\chi^{\prime}(G) \sim D(G)$ and $\phi(G) \sim D(G)$, where $\chi^{\prime}$ stands for the chromatic index and $\phi$ denote the largest number of coverings into which the edges of $G$ can be partitioned. In particular, this implies that asymptotically a.s. there is a perfect matching. Note that taking $k=2$ we get that we need $p(n) \gg \log n/n$, which is a well-known result by Erdos and Renyi.

Let $H = G_k(n,p)$ and let $p = p(n)$ be such that $p(n) = o(1)$ and $\frac{\log n}{n^{k-1}} = o(p(n))$. I would appreciate if someone could tell me where can I find a proof of $D(H) \sim d(H)$ and $C(H) = o(D(H))$ or give me some explanations. How do we study the behavior of the co-degree?

Thanks a lot